microstructure evolution in deformed polycrystals

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Microstructure evolution in deformed polycrystalspredicted by a diffuse interface Cosserat approach

Anna Ask, Samuel Forest, Benoit Appolaire, Kais Ammar

To cite this version:Anna Ask, Samuel Forest, Benoit Appolaire, Kais Ammar. Microstructure evolution in deformedpolycrystals predicted by a diffuse interface Cosserat approach. Advanced Modeling and Simulationin Engineering Sciences, SpringerOpen, 2020, 7 (1), 10.1186/s40323-020-00146-5. hal-02494630

https://hal.archives-ouvertes.fr/hal-02494630

https://hal.archives-ouvertes.fr

Ask et al.

RESEARCH

Microstructure evolution in deformed polycrystalspredicted by a diffuse interface Cosserat approachAnna Ask1*, Samuel Forest2, Benoıt Appolaire3 and Kais Ammar2

*Correspondence:

[email protected] Materiaux et

Structures, Onera, BP 72, 92322

Chatillon CEDEX, France

Full list of author information is

available at the end of the article

Abstract

Formulating appropriate simulation models that capture the microstructureevolution at the mesoscale in metals undergoing thermomechanical treatments isa formidable task. In this work, an approach combining higher-order dislocationdensity based crystal plasticity with a phase-field model is used to predictmicrostructure evolution in deformed polycrystals. This approach allows to modelthe heterogeneous reorientation of the crystal lattice due to viscoplasticdeformation and the reorientation due to migrating grain boundaries. The modelis used to study the effect of strain localization in subgrain boundary formationand grain boundary migration due to stored dislocation densities. It isdemonstrated that both these phenomena are inherently captured by the coupledapproach.

Keywords: Cosserat crystal plasticity; phase-field method; diffuse interfaces;dislocation density; grain boundary migration

IntroductionA metallic polycrystal exposed to thermomechanical treatment can undergo sig-

nificant microstructural evolution due to viscoplastic deformation and subsequent

or concurrent recrystallization processes. Heterogeneous reorientation of the crystal

lattice during plastic deformation may for instance lead to fragmentation of existing

grains. In particular, material induced strain localization (slip bands) and curva-

ture localization (kink bands) [1, 2, 3] may be associated with the formation of new

(sub-)grain boundaries [4]. The associated lattice curvature is accommodated by

geometrically necessary dislocations (GNDs). At the same time, statistically stored

dislocations (SSDs) accumulate in the microstructure due to random trapping [5].

The associated stored energy is an important driving force for grain boundary mi-

gration [6, 7, 8, 9], and in the recrystallization process it is observed that dislocation-

free nuclei form which expand into and replace grains of high stored energy content.

Detailed continuum descriptions of grain boundaries can also be based on decompo-

sition into dislocation/disclination distributions [10] or micromechanical concepts

allowing for shear-coupled boundary migration [11]. These latter specific mecha-

nisms are not addressed in the present work.

Viscoplastic deformation at the mesoscale (defined here as the grain scale of

the polycrystal) including the heterogeneous reorientation discussed above can be

successfully modeled by crystal plasticity approaches. In order to create extended

frameworks which also take into account grain boundary migration after deforma-

tion, and in some cases nucleation as an intermediate step, several authors have

mailto:[email protected]

Ask et al. Page 2 of 29

coupled crystal plasticity models with methods such as cellular automata [12, 13],

level-sets [14, 15] or phase-field methods [16, 17, 18, 19, 20] (these references are

by no means a complete list but should rather be considered a sample of works

for each mentioned approach). There are only a few works that consider a strong

coupling, formulating and solving the complete set of equations as a monolithic

system [21, 22, 23]. One of the major issues to resolve in a fully coupled framework

is the fact that crystal orientation may change through deformation or through

grain boundary migration. Often, the models that are combined rely on different

descriptors of the microstructure and it is necessary to translate between those in

intermediate processing steps.

Kobayashi, Warren and Carter (KWC) [24, 25] proposed a model that treats

the crystal orientation as a phase-field with diffuse interfaces. Their formulation

couples the evolution of the orientation phase-field to the evolution of a second, more

classical phase-field variable. In this way, the model provides direct access to the

change in crystal orientation that is due to atomic reshuffling processes in the wake

of a migrating grain boundary. The energy density that KWC introduced only takes

the norm of the orientation gradient (i.e., the lattice curvature) as a variable, and

not the orientation itself, in order to respect the requirement of frame invariance.

In particular, the inclusion of a linear term is necessary in order to produce a

solution with regions of constant orientation (grains) separated by localized regions

of non-zero curvature (grain boundaries). It is worth stressing that [26, 27] have

amended this model with a contribution due to statistically stored dislocations, so

as to address stored energy driven grain boundary migration that is the dominant

growth phenomenon during recrystallization.

In this paper, a fully coupled, thermodynamically consistent model that com-

bines a Cosserat single crystal plasticity model with the modified KWC orientation

phase-field model of [26] is used to predict microstructure evolution in deformed

polycrystals. The model was first developed for small deformations [28] and later

for large deformations in [22]. In this work, a linearized version of the large deforma-

tion model (c.f., [21]) is used. The Cosserat continuum is endowed with rotational

degrees of freedom that can be associated with the lattice directors of the crystal.

A strong coupling between the crystal plasticity model and the KWC orientation

phase-field model is made possible via the Cosserat degrees of freedom. The lattice

curvature parameter in the free energy density of the KWC model is replaced in

the coupled model by the curvature or wryness tensor of the Cosserat continuum

[29, 30].

Because the KWC model represents the lattice orientation as a continuous field

rather than relying on a cellular description of the grains, it can accommodate

the heterogeneous reorientation associated with plastic slip processes. Moreover,

because the KWC model introduces a phase field variable, it is able to consider

reorientation by a moving front. In the coupled model adopted in this work, the

grain boundary migration is not accommodated by plastic slip processes, unlike the

model in [23], where the KWC model is combined with strain gradient plasticity.

Indeed, the driving forces for grain boundary migration are capillary forces or stored

energy due to scalar dislocation densities which accumulate or recover following a

modified Kocks-Mecking-Teodosiu evolution law [26, 27, 28].


This paper is organized as follows. The diffuse interface framework is summarized

and the constitutive model is presented. A subsection is dedicated to the identifica-

tion of model parameters. Then follows the numerical examples, with a subsection

dedicated to the formation of (sub-)grain boundaries during deformation and a

subsection dedicated to grain boundary migration. Some first results toward recrys-

tallization are also presented. The paper is then concluded with a summary of the

main results and some points on future work.

Notation

Vectors will be denoted by underline as a and second order tensors by an underscore

tilde symbol as A∼ , with the transpose A∼T . Third order tensors will in the same vein

be denoted by a∼

and fourth order tensors by C≈

. Double contraction is written as

A∼ : B∼ and simple contraction as b = A∼ · a. The scalar product of two vectors can

be written as a ·b = aT b, whereas the dyadic product is written as A∼ = a∼⊗b∼. The

gradient and divergence are written by products using the nabla symbol as a ⊗∇and ∇ · a or A∼ · ∇, respectively, with differentiation of a tensor assumed to act on

the second index. The curl operator is denoted ∇×A∼ .

The second order identity tensor will be denoted I∼. The transformations between

skew-symmetric tensors and pseudo-vectors can be written by means of the Levi-

Civita permutation tensor ε∼

as

×A = −1

2ε∼

: skew(A∼ ) = −1

2ε∼

: A∼ , (1)

and

skew(A∼ ) = −ε∼·×A , (2)

respectively.

The diffuse interface Cosserat frameworkThe small deformation theory presented in this section recalls the formulation pro-

posed in [21], and represents the linearized version of the large deformation theory

developed in [22].

Governing equations

A Cosserat medium is endowed with three additional degrees of freedom at each

material point, allowing for a microrotation which is, a priori, independent of the

displacement degrees of freedom. The additional degrees of freedom necessary to

describe the microrotation can be collected in the pseudo-vector Θ. In the small

deformation setting the Cosserat rotation tensor is then given by

R∼ = I∼ − ε∼ ·Θ . (3)

The linearized, objective deformation measures are given by [29, 30]

e∼ = u⊗∇+ ε∼·Θ , κ∼ = Θ ⊗∇ . (4)


where u is the displacement vector. The curvature or wryness tensor κ∼ = Θ ⊗ ∇can be directly related to the tensor of geometrically necessary dislocation densities

α∼ through a famous result by Nye [31]

α∼ = κ∼T − tr(κ∼) I∼ , κ∼ = α∼

T − 1

2tr(α∼) I∼ . (5)

The derivation of the above expression is detailed in [30].

In the framework of small deformations the Cosserat deformation tensor is de-

composed into elastic and plastic parts as

e∼ = e∼e + e∼

p . (6)

So far, a similar decomposition of the curvature tensor has not been considered in

the coupled theory although this would be possible to introduce as outlined in [32].

In order to allow for microstructure evolution due to grain boundary migration,

[28, 22] coupled the Cosserat model to the orientation phase-field model proposed

by [24, 25]. To this end, a phase-field φ ∈ [0, 1] is introduced as an additional degree

of freedom. The phase-field variable is interpreted in the coupled model as a coarse-

grained measure of crystalline order. In the bulk of an undeformed grain the order

parameter takes the value φ = 1 whereas φ < 1 in grain boundaries. In deformed

grains the build-up of so–called statistically stored dislocations may also result in

the phase-field variable taking values φ < 1. This will be discussed in more detail

when the constitutive model is presented. Note that the crystal orientation, which

was included as a scalar phase-field in the model proposed by [24, 25], is associated

with the Cosserat degrees of freedom in the model presented here.

The derivation of the governing balance equations and boundary conditions re-

lies on the principle of virtual power and is detailed in [28, 22]. The phase-field

contribution is accounted for in the principle of virtual power by the microstress

formalism introduced by [33, 34]. There are therefore in total four work-conjugate

pairs φ : πφ,∇φ : ξφ, e∼ : σ∼ ,κ∼ : m∼ and the principle of virtual power, after some

manipulations, takes the form

∫Dφ[∇ · ξ

φ+ πφ + πext

φ

]dV +

∫∂D

φ[πcφ − ξφ · n

]dS

+

∫Du ·[σ∼ · ∇+ f ext

]dV +

∫∂Du ·[f c − σ∼ · n

]dS

+

∫DΘ ·

[m∼ · ∇+ 2

×σ + cext

]dV +

∫∂DΘ ·

[cc −m∼ · n

]dS = 0 ,

(7)

over any region D of the body Ω with outward unit normal n on the boundary

∂D. Superscripts 〈•〉ext and 〈•〉c denote external body and contact forces and cou-

ples, respectively. It follows that the balance equations and corresponding boundary


conditions for the coupled formulation are given by [28]

∇ · ξφ

+ πφ + πextφ = 0 in Ω , (8)

σ∼ · ∇+ f ext = 0 in Ω , (9)

m∼ · ∇+ 2×σ + cext = 0 in Ω , (10)

ξφ· n = πcφ on ∂Ω , (11)

σ∼ · n = f c on ∂Ω , (12)

m∼ · n = cc on ∂Ω , (13)

where n is the outward normal to the boundary ∂Ω of the body Ω. The first line

(8) gives the balance law for the microstresses related to the phase field φ and its

gradient ∇φ. It should be noted that the stress σ∼ associated with the Cosserat

deformation e∼ is not symmetric, and must therefore not be interpreted as the usual

Cauchy stress. Its skew-symmetric contribution appears also in the balance equation

for the couple-stress m∼ , which in turn is associated with the wryness tensor κ∼. In

this paper, the body forces πextφ ,f ext and cext are assumed to vanish.

Applying the standard thermodynamical principles on a free energy density of

the format ρΨ = ψ(φ,∇φ, e∼e,κ∼, rα), where rα are internal variables related to

the inelastic behavior, results in the following expression of the Clausius-Duhem

inequality [21]:

− ρ Ψ− πφ φ+ ξφ∇φ+ σ∼ : e∼ +m∼ : κ∼ =

−[πφ +

∂ψ

∂φ

]φ+

[ξφ− ∂ψ

∂∇φ

]· ∇φ+

[σ∼ −

∂ψ

∂e∼e

]: e∼

e +

[m∼ −

∂ψ

∂κ∼

]: κ∼

+ σ∼ : e∼p −

∑α

∂ψ

∂rαrα ≥ 0 ,

(14)

assuming isothermal conditions. The constitutive relations are deduced from the

energetic contribution to inequality (14):

πeqφ = −∂ψ∂φ

, (15)

ξφ

=∂ψ

∂∇φ, (16)

σ∼ =∂ψ

∂e∼e, (17)

m∼ =∂ψ

∂κ∼. (18)

Furthermore, a thermodynamic force associated with the internal variables rα can

be introduced as

Rα =∂ψ

∂rα. (19)

In order to recover the phase-field dynamics for the variable φ, it is assumed in line

with [34, 35] that the stress πφ contains stored and dissipative contributions so that


πφ = πeqφ + πneqφ , (20)

with only the stored part given by (15).

Evolution equations governing the dissipative processes are assumed to be deriv-

able from a potential (with the appropriate convexity properties to ensure non-

negative dissipation)

Ω = Ωp(σ∼) +Ωα(Rα) +Ωφ(πneqφ ) , (21)

so that

e∼p =

∂Ωp

∂σ∼, rα = −∂Ω

α

∂Rα, φ = − ∂Ωφ

∂πneqφ

. (22)

Link between the lattice rotation and the Cosserat microrotation

The rate of the Cosserat deformation can be written as

e∼ = ε∼ + ω∼ + ε∼· Θ , (23)

where the symmetric contribution is the rate of the usual small strain tensor ε∼ =12 [u⊗∇+∇⊗ u ], and ω∼ = 1

2 [ u⊗∇−∇⊗ u ] is the spin tensor. The skew-

symmetric part of the above expression can be collected in the pseudo-vector

×e =

×ω − Θ . (24)

Equation (24) gives the relation between the relative rotation of the material and

the Cosserat microrotation of a triad of microstructural directors attached to the

material point.

By defining the elastic and plastic spin pseudo-vectors×ω p :=

×e p and

×ω e :=

×ω − ×ω p, respectively, the rate of the skew-symmetric deformation (24) can now be

expressed as

×ω e − Θ =

×e e . (25)

The above equation provides the necessary link between the rotation of the crystal

lattice vectors and the Cosserat directors that is an essential part of the modeling

framework. They can be made to remain parallel by enforcing the internal constraint

×e e ≡ 0 . (26)

The Cosserat crystal plasticity framework thereby allows for a direct access to

the heterogeneous evolution of the crystal orientation in the polycrystal during

deformation.


Constitutive model and identification of parametersIn this section the specific constitutive choices are presented and the identification

of model parameters is described. Pure copper is used as a model material. The

resulting model is a non-local Cosserat crystal plasticity model with diffuse and

mobile grain boundaries. Grain boundary migration is either driven by the interface

curvature or by the energy stored during plastic deformation.

Constitutive model

The free energy density for the coupled problem is chosen as [28]

ψ(φ,∇φ, e∼e,κ∼, r

α) = f0

[f(φ) +

a2

2|∇φ|2 + s g(φ)||κ∼||+

ε2

2h(φ)||κ∼||

2

]+

1

2ε∼e : E

≈s : ε∼

e + 2µc×e e · ×e e + ψρ(φ, r

α) ,

(27)

with

ψρ(φ, rα) = φ

N∑α=1

1

2µ rα2 , (28)

where N is the number of crystal slip systems and rα are internal variables. This

contribution is taken to represent stored energy due to random trapping of dislo-

cations during plastic deformation by associating the internal variables with the

statistically stored dislocation densities, here denoted ρα, according to

rα = b

√√√√ N∑β=1

hαβρβ . (29)

Inserted into equation (28), this then gives the following expression for the energy

contribution related to the scalar dislocation densities ρα:

ψρ(φ, rα) =

1

2µ b2 φ

N∑α=1

N∑β=1

hαβρβ . (30)

This stored energy is an important driving force for grain boundary migration [7, 8].

The coupling with the phase-field variable by means of the linear term in φ ensures

that local gradients in the stored energy result in grain boundary migration [26, 27].

In the grain boundary regions, φ tends toward zero. This means that energy is

stored mainly in the bulk, and not inside the grain boundaries where the concept

of statistically stored dislocations loses its significance.

Whereas the contribution due to the phase-field φ in the energy density (27) con-

tains standard bulk and interphase terms, the contribution due to the curvature,

which can be considered as a generalization to three dimensions of the energy contri-

bution proposed by [24, 25] for the orientation phase-field, must contain the rank one

term ||∇κ∼||. It is this term that ensures a solution with localized grain boundaries.

At zero lattice curvature this term renders the energy density non differentiable

and in the numerical treatment a regularization is therefore applied [36, 28]. In the


context of non-local crystal plasticity, several authors have considered energy den-

sities that contain linear contributions due to the GND density tensor [37, 38, 39],

interpreting the linear term to represent the self energy of the GNDs whereas the

quadratic term represents the energy due to their interactions. It may be noted

that [40] found solutions with substructure partitioning, reminiscent of subgrains,

even without a rank one term for a large deformation Cosserat crystal plasticity

model. While those results are not directly applicable to the coupled model, it may

nevertheless be warranted to reconsider the optimal format of the energy density in

the future.

The functions g(φ) and h(φ) are required to be non-negative. A simple quadratic

function is chosen for the latter such that h(φ) = φ2 whereas, following [24], a

logarithmic function is chosen for g(φ) such that

g(φ) = −2 [ log(1− φ) + φ ] . (31)

This choice allows to fit a Read-Shockley type behavior for low angle grain bound-

aries. Due to the singularity for φ = 1, a small positive constant is added in the

logarithm in the numerical treatment.

In the second line in (27), the symmetric elastic deformation is included via a stan-

dard quadratic form, where E≈s is the usual elasticity tensor. The skew-symmetric

elastic deformation is penalized by choosing the Cosserat parameter µc sufficiently

large and thereby approaching the constraint (26) that the Cosserat microrotation

should follow the lattice reorientation [30, 41, 42, 43, 40]. This constraint could

alternatively be directly imposed by using Lagrange multipliers.

The stress, couple stress and microstresses take the following forms:

sym(σ∼) =E≈s : ε∼

e , (32)

×σ = 2µc

×ee (33)

m∼ = f0

[s g(φ)

1

||κ∼||+ ε2 h(φ)

]κ∼ , (34)

πeqφ = − f0[

1− φ− s ∂g∂φ||κ∼|| −

ε2

2

∂h

∂φ||κ∼||

2

]−

N∑α=1

1

2µ rα2 , (35)

ξφ

= f0 a2∇φ . (36)

From (33) together with (24), it is apparent that a migrating grain boundary,

which locally reorients the lattice as a result of atomic reshuffling processes, would

give rise to a skew-symmetric stress. In order to relax this stress and allow the

grain boundaries to migrate (without storing elastic energy), the plastic dissipation

potential Ωp is assumed to consist of two terms:

Ωp =

N∑α=1

Kv

n+ 1

⟨|τα| −Rα/φ

Kv

⟩n+1

+1

2τ−1? (φ,∇φ,κ∼)

×σ · ×σ , (37)

where 〈•〉 = Max(•, 0) and×σ contains the skew-symmetric contributions to the

stress. The choice (28) results in a Taylor type hardening law for the associated


force Rα

Rα =∂ψ

∂rα= φ µ rα = φ µ b

√√√√ N∑β=1

hαβρβ , (38)

which is interpreted as the critical resolved shear stress on slip system α. Equation

(37) has been constructed so that the value of the phase-field φ will not influence

the plastic slip flow evolution.

The first term of (37) leads to a classic crystal flow rule. The resolved shear stress

τα on slip system α can be calculated as

τα = `α · σ∼ · nα , (39)

where `α and nα are respectively the slip direction and normal to the slip plane.

Note that the additional contribution to the resolved shear stress due to the non-

symmetry of the stress tensor can be interpreted as a size-dependent kinematic

hardening [44, 4, 45, 46]. The evolution of plastic deformation follows from (22)

and (37)

e∼p =

N∑α=1

γα `α ⊗ nα + τ−1? (φ,∇φ,κ∼) skew(σ∼ ) , (40)

with the slip rate given by

γα =

⟨|τα| −Rα/φ

Kv

⟩nsign τα . (41)

The second term in (40) allows the relaxation of the skew-symmetric stress by acting

on the plastic spin ω∼p. The viscosity type term τ?(φ,∇φ,κ∼) is constructed so that it

is small in the interface region but large in the bulk of the grains, thereby restricting

the relaxation to the grain boundaries.

There are therefore two contributions to the plastic spin

×ω? = τ−1? (φ,∇φ,κ∼)

×σ , (42)

ω∼p − ω∼

? = skew(

N∑α=1

γα `α ⊗ nα ) , (43)

due to the relaxation of the skew-symmetric stress at moving interfaces and to plas-

tic slip processes, respectively. The skew-symmetric plastic deformation, likewise,

can be considered to consist of two separate contributions

×ep =

×eslip +

×e? , (44)

with

×eslip =

×ωp − ×ω? , ×

e? =×ω? . (45)


The elastic skew-symmetric deformation is then given by

×ee =

×ϑ− ×eslip − ×e? −Θ , (46)

where×ϑ is the pseudo-vector of the skew-symmetric tensor ϑ∼ = skew(u ⊗ ∇ )

denoting the material rotation.

In the present model, the Cosserat microrotation is associated with the lattice

(crystal) orientation as measured with respect to some fixed frame. It is therefore

in general non-zero in the reference configuration. It follows [28, 21] that the ini-

tial plastic deformation must be non-zero for the reference configuration to remain

stress-free. Specifically, this is obtained by adopting the initial condition

×ep(t = 0) =

×e?(t = 0) = −Θ(t = 0) . (47)

The part of the inelastic deformation that is not due to slip processes during de-

formation can therefore be seen as necessary to accommodate an initial orienta-

tion distribution created from a previous, homogeneous state. It can be interpreted

therefore, somewhat simplified, as representing a reference orientation state of the

grain, and its associated evolution law ensures that it is inherited to a region swept

by a migrating grain boundary. The evolution of the plastic deformation due to

migrating grain boundaries according to equation (45) was reported in [22].

The standard evolution equation is obtained for the phase-field φ by choosing a

quadratic dissipation potential [33, 26]

Ωφ =1

2τ−1φ

(πneqφ

)2 , (48)

where τφ is a viscosity type parameter, so that

τφ φ = −πneqφ . (49)

Instead of formulating a dissipation potential for the internal variables, which

are associated with the scalar dislocation density in equation (29), modified Kocks-

Mecking-Teodosiu evolution laws are used, given by

ρα =

1b

(1K

√∑β ρ

β − 2dρα)|γα| − ρα CD A(|κ∼|) φ if φ > 0

1b

(1K

√∑β ρ

β − 2dρα)|γα| if φ ≤ 0

(50)

The additional recovery term, with parameter CD, was originally introduced by

[26, 27]. It allows for static recovery behind the front of a migrating grain boundary,

with the amount of recovery governed by the parameter CD. The static recovery is

localized to the interface region by the function A(|κ∼|) = tanh(C2A ||κ∼||2). The com-

peting nucleation and annihilation mechanisms in equation (50) determine whether

energy is stored or dissipated. In particular, the additional recovery term allows for

stored energy to be released through grain boundary migration.


Table 1 Model parameters used in the simulations.

Parameter value unit commentElasto-viscoplastic parameters

C11 160 GPa Elasticity moduli for cubic symmetryC12 110 GPaC44 75 GPaµc/f0 11.5 GPa Cosserat modulusµ 46 GPa Mean shear modulusb 0.2556 nm Burgers vector (magnitude)Kv 10 MPa s1/n Viscoplasticity parametersn 10K 1/10 Kocks-Mecking parametersd 10 nmCD 100 Determines the rate of static recoveryCA 1 µm

Phase-field and mobility parametersf0 1.15 MPa Sets the magnitude of the grain boundary energys 1.5 µma 0.62 µmε 2 µm Sets the grain boundary width

τφ/f0 100 s Viscosity type (mobility) parameter for φ

τ?/f0 10 s Viscosity type parameter for×ω?

Identification of model parameters

The model parameters used in the simulations are given in table 1. While typical

values for most of the elasto-viscoplastic material parameters of pure copper at

room temperature can be found in the literature (in this paper values adapted from

[47, 48] are used), table 1 also gives values for the recovery parameter CD in (50)

and the Cosserat coupling modulus. The former is chosen so that full static recovery

takes place in the wake of migrating grain boundaries. For the moment, only self

interaction between static dislocations is considered. In equation (29), hαβ = 0.3 if

α = β and otherwise hαβ = 0 if α 6= β. The Cosserat modulus is chosen sufficiently

large to act as a numerical penalty parameter.

The parameters f0, a, and s, related to the phase-field inspired energy contribution

in (27), are calibrated to fit values obtained from atomistic calculations for < 100 >

tilt grain boundaries. Following [28], the general calibration procedure relies on the

matched asymptotic analysis of the original KWC model as proposed in [49]. At the

lowest order corresponding to equilibrium, the coupled model becomes equivalent

to the KWC model. Hence, using the results from [49], it is possible to calculate

the grain boundary energy as a function of the misorientation angle. Some grain

boundary energies – calculated by atomistic simulations [50, 51, 52] for copper

tilt grain boundaries – are collected in figure 1 (top). The grain boundary energy

exhibits cusps, i.e., favored misorientations which result in local energy minima. The

current model does not take into consideration such local minima so the calibration

was only performed on the first part of the curve up to a misorientation angle of 30.

The result is shown with stars in figure 1 (bottom) and the calibrated parameters

are given in table 1. The parameter ε controls (together with a) the width of the

grain boundary. It is chosen to be ε = 2 µm, which results in a diffuse interface

width of around 500 nm. The interface width puts a restriction on the element size

in the finite element discretization. At least four to five nodes, but ideally more, are

required to resolve the curvature.

Whereas the coupled Cosserat model and the model proposed in [26] (which is

close to the original KWC model) become similar at equilibrium, they differ in dy-


Figure 1 Top: Grain boundary energy as a function of misorientation for copper, obtained fromatomistic calculations [50, 51, 52]. Bottom: Result from calibration of model parameters for theorientation phase-field model (black stars) to fit the calculated grain boundary energy of a< 100 > tilt grain boundary from [50] (solid blue line). Only angles up to 30 are considered inthe calibration because the model does not capture the cusps seen in the full range ofmisorientation angles.

namic situations. The two formulations have been compared in [28, 21]. The model

in [26] relies on a relaxation dynamics for the orientation, whereas in the coupled

model the orientation is at each instant governed by the constraint (46) together

with the balance equation (34). Because of this difference, it is not possible to use

the asymptotic analysis of the KWC model to calibrate the mobility parameters of

the coupled model, because only the expansion to lowest order is valid in this case.

The choice of the parameter τ?(φ,∇φ,κ∼) in equation (42) was studied in [28]

where it was shown that it must be bounded by the choice of τφ in order not to

restrict the grain boundary mobility. It will be assumed to be of the general form

τ?(φ,∇φ,κ∼) = τ? P?(φ, ||κ∼||) . (51)

The function P?(φ, ||κ∼||), which is meant to localize the stress relaxation to the

interface region, will be discussed in the next section. In concordance with [28], the

parameter τ? is chosen to be τφ/10. Both values are given (scaled by f0 for clarity)

in table 1. The choice of the parameters for the dynamic behavior is revisited in the

next section when the numerical simulations are discussed. A formal procedure for

identifying the dynamic coefficients of the model remains to be established, which


is why in this work the mobility parameter τφ is adjusted by running numerical

simulations on simplified test cases.

Microstructure evolution in deformed polycrystalsIn this section, the microstructure evolution predicted by the proposed model in

deformed polycrystals is investigated by numerical simulations. In particular, the

effect of localized reorientation of the lattice and the grain boundary migration due

to statistically stored dislocation densities are studied. The numerical simulations

are performed using the finite element code Z–set [53]. The discretization of the weak

form of the balance equations results in a monolithic, coupled system of equations

which is solved using implicit integration in an iterative Newton-Raphson scheme.

The constitutive equations are integrated using a fourth order Runge-Kutta method

with automatic time-stepping. For details on the numerical implementation, see

[28, 22] and also [41] and [26] for the Cosserat crystal plasticity and the phase-field

model, respectively. Since only planar problems are considered, Θ = [ 0 0 θ ]T ,

with x3 considered to be the out-of-plane axis and θ3 = θ the only non-zero ori-

entation degree of freedom. The planar problem therefore has only four degrees of

freedom: two components of displacement, the microrotation θ, associated with the

crystal orientation, and the phase-field φ.

Periodic geometries and boundary conditions are considered. Simple shear loading

is then applied by imposing

u = B∼ · x+ p (52)

with the tensor B∼ given by

B∼ =

0 B12 0

B21 0 0

0 0 0

, (53)

with only either B12 or B21 non-zero. The vector p is a periodic fluctuation that

takes the same value in opposite points of the boundary.

Grain boundary mobility

As discussed in the previous section, the grain boundary mobility cannot be cali-

brated based on results for the orientation phase-field model, as the evolution equa-

tions for the two models are different. While a formal calibration procedure remains

to be established, the grain boundary mobility can be estimated and adjusted by

running a simple test case. This test case is a periodic bigrain of the type considered

in [28], with a flat grain boundary and 15 misorientation between the two grains.

One grain is assigned a (constant) scalar dislocation density and the other grain

is assumed to be dislocation-free. The resulting stored energy gradient acts as a

driving force for grain boundary migration. With a planar grain boundary and a

constant stored energy gradient, the grain boundary mobility can be calculated as

M = v/ψρ, with v being the grain boundary velocity and ψρ the pressure acting

on the grain boundary due to the stored energy.


In the viscosity type parameter τ?(φ,∇φ,κ∼) in equation (51), the localizing func-

tion is chosen to be

P?(||κ∼||) = 1−[ µpε− 1

]exp(−βp ε ||κ∼||) , (54)

following [25]. In [28], the same localizing function was used for the orientation

phase-field, whereas a tanh function was used for the τ? coefficient. The constant

µp determines the value of P?(||κ∼||) in the bulk of the grains and should be chosen

to be large so that 1/τ? → 0 there. The constant βp sets the width of the region

around the grain boundary where P?(||κ∼||) ≈ 1 and should be chosen βp ≤ 103 in

order to achieve localization. In the simulations, µp = 109 µm and βp = 102 are

used.

With the driving pressure due to the stored dislocation density given by ψρ = 0.23

MPa and using the material parameters given in table 1, the resulting simulated

grain boundary velocity is v = 0.8 nm/s. The resulting mobility is M = 3.5

nm/(MPa s). While there is not a plethora of available experimental data on the

grain boundary mobility, [54] measured an average grain boundary mobility of 0.61

nm/(MPa s) at 121C for recrystallization of cold deformed copper. Theoretical

average grain boundary mobilities were also calculated in [54]. Compared to that

data,M = 3.5 nm/(MPa s) corresponds to recrystallization at a temperature of ap-

proximately 150-160C. In the present work, thermal effects are not accounted for

in the simulations, and the indicated temperature range should not be considered

predictive of the true final temperature of the system.

In order to compare the behavior of the coupled model and the orientation phase-

field model, the latter was also used to run the same example of a periodic bi-

grain with a stored energy gradient. The model parameters, where applicable, were

the same as for the coupled model, and in the evolution equation for the orienta-

tion phase-field (c.f. [24, 26, 28]), the mobility parameter was chosen identical to

τ?(φ,∇φ,κ∼). The mobility predicted by that simulation wasM = 0.87 nm/(MPa s),

which is lower than for the coupled model. This is as expected and further indicates

that it is necessary to establish a separate calibration procedure for the mobility of

the coupled model.

Initial conditions for polycrystal simulations

In general, initial microstructures for polycrystalline aggregates are obtained from

measurements of real samples, or generated by tessellation using dedicated software

such as Neper [55]. The tessellations can then be translated into a finite element

mesh with discrete regions of different orientation, i.e., grains. In this work, two

periodic, planar microstructures generated with Neper are used, one with six grains

and one with 32 grains. The smaller geometry is 20 × 20 µm in size, discretized

with 8464 (92 × 92) elements with quadratic interpolation functions and reduced

integration. The larger geometry is 50 × 50 µm, and it is discretized with 47089

(217×217) elements. The element size is therefore nearly the same in both examples,

0.21 µm in the six grains geometry compared to 0.23 µm in the one with 32

grains. The orientations are distributed in the range 0 to 35, which corresponds

to the range of misorientations in the grain boundary energy for which the model

is calibrated.


Figure 2 Initial orientation distribution generated with Neper (left) and the orientationdistribution after initial simulations with the orientation phase-field model (right). The orientationis given in radians. Because the orientation behaves as a phase-field, the grain boundaries in thecoupled model are not restricted to follow the finite element grid and so there is no need to adaptthe mesh to the morphology of the microstructure.

The grain morphologies and orientation distributions obtained using Neper are

not suitable for use as initial conditions in the coupled model as they are far from

the equilibrium solution, with jagged edges tied to the finite element grid, and so

will result in convergence problems (figure 2, left). In order to generate suitable

initial conditions for the non-zero component θ of×Θ as well as φ, simulations using

the KWC orientation phase-field model are performed. This will also generate initial

conditions for the non-zero component×ep of

×ep (c.f., equation (47)). The simulations

take the grain morphologies generated by Neper as initial conditions, together with

a uniform φ = 1. The simulations are run until a solution with localized, smooth

grain boundaries is found. Figure 2 shows the microstructure generated by Neper

for the geometry with six grains and the corresponding output for the orientation

field from the phase-field simulation. This field will serve as initial conditions on θ

and -×ep in the coupled simulations, as shown in figure 3 (top left). The phase-field

simulations also find the corresponding solution for φ, shown in figure 3 (bottom

left). Initial conditions for the geometry with 32 grains are generated by the same

procedure. In all simulations, the dislocation densities have a uniform initial value

of ρα = 2 · 1011 m−2.

Kink bands and subgrain formation

Up to four possible slip systems are considered for the planar problem with the slip

directions given by

l1 = ( 1, 0 ) , l2 = ( 0, 1 )

l3 =1√2

( 1, 1 ) , l4 =1√2

(−1, 1 ) .(55)

The slip plane normal in each case is perpendicular to the slip direction. By choosing

which slip systems are allowed to be active in a simulation, localization phenomena

such as kink or slip bands can be triggered. In particular, the formation of kink

bands, which are bands of localized plastic deformation that are perpendicular to

the slip direction, can result in large localized reorientation of the lattice (see the

recent contribution [3] for a detailed description of kink and slip bands in crystals


Figure 3 The six grain microstructure before and after applied shear deformation. The initialvalues are shown to the left for θ (top), φ (middle) and norm of the curvature |∇θ| (bottom). Topright shows the orientation distribution after shear deformation, middle right shows thecorresponding phase-field φ and bottom right shows the norm of the curvature. The truemaximum value of the curvature norm is around |∇θ| = 1.9 µm−1 but it has been capped off at1 µm−1 for a better visualization. In the simulation, only one slip system α = 2 was active,resulting in the formation of kink bands and subgrains. The slip direction of each grain isindicated with black lines in the top right figure.

and how they arise in crystal plasticity simulations). In the simulations, a periodic

shear deformation is applied with B21 = 0.05 in equation (52). Kink bands are

triggered when only one slip system can be activated.

Figure 3 shows the orientation field θ (top), the phase-field φ (middle) and the

curvature norm |∇θ| (bottom) before (left) and after (right) deformation of the six

grains microstructure. The deformation is not shown, i.e., the fields are projected

onto the undeformed mesh (the total shear deformation is five percent). Only slip

system α = 2 was allowed to activate in the simulation. The formation of kink

bands is most easily seen in the bottom right figure, which shows the norm of the

curvature |∇θ| in the deformed structure. These bands are also visible in the phase-

field φ. Note that the grains are oriented between 0 − 35 from the x1-direction,

so the localized bands are perpendicular to the slip direction since it is the (0, 1)

slip direction that is active. In the top left image, the slip direction of each grain


Figure 4 The six grain microstructure before and after applied shear deformation. The initialvalues are shown to the left for θ (top), φ (middle) and norm of the curvature |∇θ| (bottom). Topright shows the orientation distribution after shear deformation, middle right shows thecorresponding phase-field φ and bottom right shows the norm of the curvature. The truemaximum value of the curvature norm is around |∇θ| = 1.9 µm−1 but it has been capped off at1 µm−1 for a better visualization. In the simulation, only one slip system α = 1 was active,resulting in the formation of kink bands. The localization is less pronounced than in the casewhere only slip system α = 2 was active. The slip direction of each grain is indicated with blacklines in the top right figure.

is indicated with black lines. The orientation phase-field method always localizes

grain boundaries where there is a sufficient orientation gradient [24], and clearly

this property is inherited to the coupled model, as may be expected. In particular,

a kink band has formed across the bottom part of the left central (green) grain and

the right central (blue) grains, resulting in what appears to be two, maybe three,

new grain boundaries delimiting new subgrains. The locations of the new grain

boundaries are near the spots marked 1 and 2, respectively, in the top right image.

Here, the term subgrain is used to denote regions that result from partitioning of the

original grain structure due to the deformation. The model automatically handles

localization and, while it is sensitive to the magnitude of the orientation gradient,

does not differentiate between different types of grain boundaries.

Figure 4 shows the results from calculations when only slip system α = 1 was

allowed to activate. Kink band formation is again observed but the localization is


Figure 5 The 32 grain microstructure before and after applied shear deformation. The initialvalues are shown to the left for θ (top), φ (middle) and curvature |∇θ| (bottom). Top right showsthe orientation distribution after shear deformation, middle right shows the correspondingphase-field φ and bottom right shows the curvature. The true maximum value of the curvaturenorm is around |∇θ| = 1.7 µm−1 but it has been capped off at 1 µm−1 for a better visualization.In the simulation, only one slip system was active, resulting in the formation of kink bands andsubgrains.

less pronounced than in the previous example. Slip band formation is not observed.

The figure shows the orientation field θ (top), the phase-field φ (middle) and the

curvature norm |∇θ| (bottom) before (left) and after (right) deformation.

Figure 5 shows the same results as figure 3, but for the larger geometry with

32 grains. Again, the formation of kink bands is most notable in the contour map

showing the norm of the orientation gradient |∇θ| in the deformed microstructure

(bottom right), but it is also visible in the phase-field φ (middle right). In particular,

below the spot marked with 1 in the top right image, a little bit above the center

of the geometry, there are two new subgrains that seem to have formed due to

the reorientation at the edges of two existing grains. The simulation shows how a

kink band associated with a strong lattice curvature transforms into a true grain

boundary, which is associated with a significant decrease in φ due to the large

accumulation of dislocation densities.


Figure 6 The accumulated plastic slip γ (left) and the scalar dislocation density ρ (right) withonly one active slip system. The maximum display value for ρ has been set to 5 · 1014 m−2 (thescale in the colorbar is in µm−2) in order to more clearly demonstrate how it is distributed overthe grains. The true maximum value is 2.4 · 1015 m−2. For the same reason, the maximum displayvalue for γ has also been limited (at 0.3). In a very narrow region larger values are observed, witha maximum of almost 1.

The interest of these examples is that by triggering kink deformation in the sim-

ulations, it is possible to study how the model behaves when very localized reorien-

tation takes place. Clearly, the coupled framework allows for the formation of new

subgrains by reorientation without the need for post-processing or reinterpretation

of the results.

Dislocation driven grain boundary migration

During the elasto-viscoplastic deformation, energy is stored in the bulk of the grains

due to random trapping of dislocations. This is included in the model via the energy

term (28) and the modified Kocks-Mecking-Teodosiu evolution law (50) for the evo-

lution of the scalar dislocation densities ρα. Because the energy contribution (28)

is coupled with φ in the energy density, a difference in stored energy between two

grains contributes to grain boundary migration. This effect is illustrated by com-

paring simulations performed with only one active slip system (the same as in the

previous example) or all four slip systems of (55). In the former case, the difference

in stored energy is higher, and correspondingly there is a stronger contribution to

the microstructure evolution due to the statistically stored dislocation densities.

Figure 6 shows the accumulated plastic strain (left) and static dislocation density

(right) in the case of only one active slip system for the six grain geometry. The

maximum scalar dislocation density is in reality 2.4 ·1015 m−2, but in the figure the

maximum display value has been limited to 5 ·1014 m−2 in order to better highlight

how the dislocation density is distributed between different grains. In figure 7, the

scaled stored energy ψρ/f0 with one (left) or four (right) active slip systems is

shown. When all four slip systems are active, energy is stored much more evenly in

the grains. The maximum value is around 0.9 for one possible slip system, whereas it

is only around 0.1 with four possible slip systems, so its contribution to the driving

force for grain boundary migration in the latter case is considerably smaller. Note

that in figure 7, the maximum display value in the left figure (one possible slip

system) has been limited to 0.5 in order to better represent how the stored energy

is distributed in the different grains.

The microstructure evolution is studied by applying a shear deformation and then

holding it constant for two hours under a moderately elevated temperature. The


Figure 7 The non-dimensionalized stored energy ψρ/f0 with one (left) or four (right) active slipsystems. The scale for the left-hand image has been cut-off at 0.5 in order to more clearly showhow the stored energy is distributed in the different grains. The true maximum value is aroundψρ/f0 = 0.9. The scale for the right-hand side image has not been altered.

grain boundary mobility was chosen to correspond with a (constant) temperature

of around 150C. This value is not predictive of the true temperature of the system;

it would be necessary to fully take into account thermal effects to achieve this. The

final microstructure for the six grains microstructure is shown in figure 8 for one

(left) or four (right) possible active slip systems. The images show, from top to

bottom, the orientation field, the phase-field φ and the curvature |∇θ|, respectively.

The initial configuration is the same for both cases and the initial grain boundaries

are outlined in black.

It is clear that the predicted grain structures are very different. With only one

active slip system, it seems that the grain boundary migration is primarily driven

by the difference in stored energy associated with statistically dislocation densities.

With four possible slip systems on the other hand, it appears that the grain bound-

ary migration is driven mostly by the grain boundary curvature. This is evident

from the shrinking of the smallest grain, which has a slightly lower stored energy

than its neighbor to the left and therefore should expand in that direction if this

had been the main driving force. Instead, it shrinks to lower the energy in the triple

junction. It is also clear from comparing the two examples that the stored energy

due to the scalar dislocation densities is a stronger driving force than curvature for

grain boundary migration in this case. This is in agreement with the typical driv-

ing pressures given in [7], with typical driving pressures due to stored dislocations

generally several magnitudes larger than those due to the grain boundary energy.

To study the evolution of the statistically stored dislocations, the simulation with

one possible active slip system is considered again. In figure 9, the dislocation density

in the grains is compared at two time points: just after the deformation is applied

(left) and after two hours (right). The grain with the highest dislocation density

post deformation has vanished completely after two hours, and in the wake of the

migrating grain boundaries, static recovery has left a region free of statistically

stored dislocations.

The larger geometry with 32 grains is also studied. Figure 10 shows, from left to

right, the orientation field, the phase-field φ and the norm of the curvature |∇θ| after

one hour of final deformation maintained constant. The initial conditions and the

corresponding fields immediately after the deformation is applied are given in figure


Figure 8 The microstructure after two hours of constant shear deformation at moderatelyelevated temperature. The simulations allow one (left) or four (right) possible active slip systems.From top to bottom the images show the orientation field, the phase-field φ and the curvature|∇θ|. The original grain boundaries are outlined in black. The initial microstructure is the same inboth cases.

Figure 9 The scalar dislocation density ρ with only one active slip systems just after deformationhas been applied (left) and two hours later (right). The maximum display value for ρ has been setto 5 · 1014 m−2 (the scale of the colorbar is in µm−2) in order to more clearly demonstrate how itis distributed over the grains. The true maximum value is 2.4 · 1015 m−2.

5. The microstructure has evolved relatively little in the time interval considered.

In part, this may be due to the distribution of the stored energy inside the grains,


Figure 10 The microstructure of the 32 grain geometry after holding the deformation constantfor one hour. Top left: the orientation field, and top right: the phase-field φ, and bottom: thenorm of the curvature |∇θ|. There is only one possible slip-system.

Figure 11 The dislocation density distribution in the 32 grain microstructure immediately afterdeformation (left) and after 35 minutes of constant deformation (right). The maximum displayvalue for ρ has been set to 5 · 1014 m−2 (the scale of the colorbar is in µm−2) in order to moreclearly demonstrate how it is distributed over the grains. The true maximum value is 1.9 · 1015m−2. There is only one possible slip system in the simulations.

which may be less favorable to grain boundary migration than in the smaller sample.

The maximum value of the stored energy is also lower than in the previous example,

approximately 1.9·1015 m−2 (in figure 11, the maximum display value for ρ has been

set to 5 · 1014 m−2). In the smaller sample, most of the microstructure evolution

takes place within the first hour, so the shorter time span considered in the larger

example is not in itself sufficient to explain the difference.

Observed phenomena for further investigation

For both considered geometries, there are some phenomena that appear in the

simulations that may require further investigation. Figure 12 shows a zoom on the

subgrain boundary that has formed in the lower left corner of the six grain geometry


Figure 12 A zoom on a new (sub-)grain boundary generated in the six grain microstructure dueto kink band formation, c.f. figure 3 and 6 (the grain boundary appears in the lower left corner ofthe microstructure in those figures). From top to bottom the images show the orientation, thedislocation density ρ, the norm of the curvature |∇θ|, the phase-field φ, and the resolved shearstress τ (ranging from -160 MPa to 230 MPa). The evolution is shown from left to right: justafter deformation and then after five minutes, one hour, and two hours of deformation. It appearsthat the reoriented bands (blue above and yellow below) are slightly fragmented, with scattereddarker spots. As the microstructure evolves, the reoriented bands start to vanish except for thesespots, which appear to remain. In the stored energy, blue regions appear at the same positions,indicating that static recovery has taken place there.

after deformation, c.f. figure 3 and 6. Only one slip system is allowed. From top to

bottom the images show the orientation, the dislocation density ρ, the norm of the

curvature |∇θ|, the phase-field φ, and the resolved shear stress τ (ranging from

-160 MPa to 230 MPa). The time evolution is shown from left to right: just after

deformation and then after five minutes, one hour, and two hours of deformation.

Already immediately after the deformation is applied, it seems that the reoriented

bands (blue above and yellow below) are not uniform and that there are small spots

of darker color dispersed inside them. Already after five minutes, and even more so

after one hour, the reoriented bands have mostly vanished (due to local reorienta-

tion) except for these spots. Furthermore, in the same position, there appears in

the scalar dislocation density small regions where static recovery has taken place


but without apparent grain boundary migration (recovery in connection with grain

boundary curvature on the other hand seems to have taken place to the right of

the region of interest, where the grain to the right has grown into the grain to the

left). There is some, but very little additional change between t = 1h and t = 2h,

and there is no definite indication for the moment that these small regions will

eventually expand.

The same behavior can be seen for the larger simulation geometry in figures 10 and

11. There are two grains a little bit above the center (below the spot marked with 1 in

figure 5, top right) where there is significant local reorientation and a similar pattern

of spots appear in the orientation field (figure 10 left) and corresponding regions of

apparent static recovery in the dislocation density (figure 11 right). In order to study

if this behavior is due to poor resolution of the localization taking place in the kink

bands, a second simulation was performed for the six grain geometry with a finer

mesh with 20449 (143 by 143) elements (the coarser mesh has 8464 elements). The

same type of behavior is again observed, as shown in figure 13. It is thus present

in two different geometries, and for the same geometry with two different finite

element sizes. Further investigation is necessary to determine the precise origin of

the irregularities in the orientation distribution, and if it is possible that they could

act as seeds for new grains and subsequent recrystallization. The evolution of the

other fields, notably the scalar dislocation density, can be explained by the time

evolution predicted by the model as a consequence of the heterogeneities in the

microstructure. The phase field φ always follows the orientation field, as indicated

in the previous examples, and this also activates the static recovery term in the

modified Kocks-Mecking-Teodosiu equations (50).

The second phenomena that may warrant further investigation appears in the

32 grain geometry, where one grain boundary exhibits signs of bulging. The grain

boundary can be located in figure 5 (top right) where it is near the spot marked

with a 2. This is shown in figures 14 and 15. The grain has a boundary near the

edge of the simulation box (in the periodic geometry), and so the change in the

profile to the very right in the figures is likely due to ordinary grain boundary

migration. However, a little to the left there is a small bulge that appears which

can not be explained by the geometry. This bulge is apparent in both the orientation

field in figure 14 and in the phase-field φ in figure 15 (top). In addition, it seems

to be associated with static recovery of the dislocation density ρ, shown in figure

15 (bottom). Again there is not much evolution however after the bulge initially

appears. It seems to stay at the same size between 1000 seconds and one hour in

figure 14. It is likely that it will be eaten up by the moving front to the right before

it evolves much further.

Summary and conclusionsMicrostructure evolution in deformed polycrystals is studied in this work by finite

element simulations, applying a constitutive model that combines Cosserat single

crystal plasticity with a phase-field model. It is shown that the model is capable of

taking into account significant microstructural changes. A major result is the impor-

tance of strain localization (slip bands) and curvature localization (kink bands) on

subsequent grain boundary formation and grain boundary migration. Simulations


Figure 13 A zoom on a subgrain boundary generated in the six grain microstructure due to kinkband formation, c.f. figure 3 and 6 (the subgrain boundary appears in the lower left corner of themicrostructure in those figures). Two solutions are compared after six minutes of constantdeformation, which is when the very localized recovery of the dislocation density starts to appear.To the left, the same mesh was used as in the previous figure whereas in the images to the right,a finer mesh was used. From top to bottom, the images show the orientation, the dislocationdensity ρ and the norm of the curvature |∇θ|. The two solutions are not identical, but the samephenomena can be observed with localization near or inside regions with high local curvature anddislocation density.

with only one possible slip system are used to trigger the formation of kink bands

and obtain curvature localization. There is some clear evidence in the results of new

grain boundaries appearing – that is to say: it is demonstrated that the phase-field

variable is sensitive to regions of high dislocation density and that the coupled sim-

ulation model has inherited the quality of the KWC orientation phase-field model

to localize grain boundaries in regions of strong curvature.

Grain boundary migration is also present in the simulation results, both due to

stored energy (SSD densities) and to curvature. The most important driving pres-

sures for grain boundary migration are obtained when large stored energy gradients

occur between individual grains.

The presented simulations are still at their early stage (2D, small strain and

rotations) but they indicate that the model contains physical ingredients that are

responsible for subgrain boundary formation, formation of possible recrystallization

seeds, and evidence of bulging phenomenon. It will be necessary in a first step, which

is a current focus, to extend the model to perform more realistic 3D simulations. The

full 3D version of the model requires seven degrees of freedom at each material point,

rather than four in the 2D case. The increase in numerical cost is therefore very

significant. The main challenge, however, is the representation of the orientation


Figure 14 A zoom on a grain boundary in the 32 grain microstructure, c.f. figure 5 and 10 (thesubgrain boundary appears in the lower right quadrant of the microstructure in those figures).From left to right, top to bottom, the orientation field around the grain boundary is shownimmediately after deformation, at 500, 1000 and 3600 seconds, respectively. The grain is on theedge of the simulation box, and so the change in the profile to the very right in the figure is likelydue to ordinary grain boundary migration. However, a little to the left there is a small bulge thatappears which can not be explained by the geometry.

Figure 15 A zoom on a grain boundary in the 32 grain microstructure, c.f. figure 5 and 10 (thesubgrain boundary appears in the lower right quadrant of the microstructure in those figures). Topline shows the phase-field φ and bottom line the dislocation density ρ. The images on the left areimmediately after the deformation is applied and the images on the right are after one hour ofconstant deformation. The maximum display value for the dislocation density has not been cappedin these images, unlike in previous figures.

field in 3D. In the plane case, the grain orientation is treated as a scalar field, with


rotation only around one major axis. The fact that the orientation is restricted to

the plane in the 2D version of the model severely limits the types of slip systems that

can be studied, which will not be the case in a 3D formulation. However, if rotation

around several axes is considered, the interpolation of the variation of orientation

between two grains requires some consideration.

The large deformation formulation of the model has already been established [22]

and in a second step, its numerical implementation will allow to study processes

such as cold working and subsequent annealing but also to tackle the more complex

problem of dynamic recrystallization with concurrent deformation, nucleation and

grain growth. This would for example allow to investigate if the model can predict a

real threshold in stored energy for strain induced grain boundary migration to take

place. While the simulations performed in the present study showed some tendency

toward grain boundary bulging, it is not clear whether a critical work-hardening

can be an outcome of the model or if the model must be enhanced.

Author’s contributions

The coupled Cosserat phase-field theory has been developed by S. Forest, A. Ask and B. Appolaire. The

implementation of the simulation model has been done by K. Ammar and A. Ask together. All simulations in this

article were run by A. Ask. B. Appolaire contributed to the asymptotic analysis which was necessary to calibrate the

model parameters. The calibration was done by A. Ask. The paper has been written primarily by A. Ask and S.

Forest with important discussions and contributions from B. Appolaire.

Acknowledgements

Not applicable.

Availability of data and materials

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Funding

This project has received partial funding from the European Research Council (ERC) under the European Union’s

Horizon 2020 research and innovation program (grant agreement n 707392 MIGRATE).

Author details1Departement Materiaux et Structures, Onera, BP 72, 92322 Chatillon CEDEX, France. 2MINES ParisTech, PSL

University, Centre des materiaux (CMAT),CNRS UMR 7633, BP 87, 91003 Evry, France. 3Institut Jean Lamour,

CNRS-Universite de Lorraine, 2 allee Andre Guinier, 54000 Nancy, France.

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